zbMATH — the first resource for mathematics

Uniqueness of self-similar solutions to the Riemann problem for the Hopf equation with complex nonlinearity. (English. Russian original) Zbl 1362.35069
Comput. Math. Math. Phys. 56, No. 7, 1355-1362 (2016); translation from Zh. Vychisl. Mat. Mat. Fiz. 56, No. 7, 1363-1370 (2016).
Summary: Solutions of the Riemann problem for a generalized Hopf equation are studied. The solutions are constructed using a sequence of non-overturning Riemann waves and shock waves with stable stationary and nonstationary structures.

35C06 Self-similar solutions to PDEs
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
Full Text: DOI
[1] Kulikovskii, A. G., A possible effect of oscillations in the structure of a discontinuity on the set of admissible discontinuities, Sov. Phys. Dokl., 29, 283-285, (1984)
[2] A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Chapman and Hall/CRC, London, 2001; Fizmatlit, Moscow, 2012). · Zbl 0926.35011
[3] Kulikovskii, A. G.; Chugainova, A. P., Simulation of the influence of small-scale dispersion processes in a continuum on the formation of large-scale phenomena, Comput. Math. Math. Phys., 44, 1062-1068, (2004) · Zbl 1136.35421
[4] Kulikovskii, A. G.; Chugainova, A. P., Classical and nonclassical discontinuities in solutions of equations of nonlinear elasticity theory, Russ. Math. Surv., 63, 283-350, (2008) · Zbl 1155.74019
[5] Chugainova, A. P., Nonstationary solutions of a generalized Korteweg-de Vries-Burgers equation, Proc. Steklov Inst. Math., 281, 204-212, (2013) · Zbl 1292.35263
[6] Chugainova, A. P.; Shargatov, V. A., Stability of nonstationary solutions of the generalized KdV-Burgers equation, Comput. Math. Math. Phys., 55, 251-263, (2015) · Zbl 1321.35196
[7] Il’ichev, A. T.; Chugainova, A. P.; Shargatov, V. A., Spectral stability of special discontinuities, Dokl. Math., 91, 347-351, (2015) · Zbl 1326.35316
[8] Chugainova, A. P.; Shargatov, V. A., Stability of discontinuity structures described by a generalized KdV-Burgers equation, Comput. Math. Math. Phys., 56, 263-277, (2016) · Zbl 1346.35178
[9] Levin, V. A.; Markov, V. V.; Osinkin, S. F., Modeling of detonation initiated in an inflammable gas mixture by an electric discharge, Khim. Fiz., 3, 611-613, (1984)
[10] Levin, V. A.; Markov, V. V.; Osinkin, S. F., Detonation wave reinitiation using a disintegrating shell, Dokl. Phys., 42, 25-27, (1997)
[11] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Nauka, Moscow, 1986; Pergamon Press, Oxford, 1987). · Zbl 0146.22405
[12] Konyukhov, A. V.; Likhachev, A. P.; Oparin, A. M.; Anisimov, S. I.; Fortov, V. E., Numerical modeling of shock-wave instability in thermodynamically nonideal media, J. Exp. Phys., 98, 811-819, (2004)
[13] Oleinik, O. A., Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation, Am. Math. Soc. Transl., Ser. 2, 33, 285-290, (1963) · Zbl 0132.33303
[14] Kulikovskii, A. G., Surfaces of discontinuity separating two perfect media of different properties: recombination waves in magnetohydrodynamics, J. Appl. Math. Mech., 32, 1145-1152, (1968)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.